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this is Rajeev Srivastava and his colleague Ravinder Padmanabha, specializing in computational geometry algorithms for application in science and research. We would like to include methods from algebraic graph theory in our research.

May we ask a conceptual question about conjugacy in algebra. In group theory, two elements $a$ and $b$ of a group are conjugate if there is an element $g$ in the group such that $b = g^{–1}ag$. This gives an equivalence relation whose equivalence classes are called conjugacy classes.

We read in the Wikipedia article that "members of the same conjugacy class cannot be distinguished by using only the group structure." What does that mean in a rigorous sense?

As an additional question, as the conjugacy classes define equivalence classes, they should generate a normal subgroup. What is the conceptual interpretation of this subgroup? Does it allow a geometric interpretation?

Thank you very much in advance.

(we included edits in our question based on the comments; thank you for your comments, for which we are very grateful)

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    $\begingroup$ I would read the informal sentence "cannot be distinguished by only using the group structure" as that there is a group isomorphism $G\rightarrow G$ sending one element to the other. That isomorphism sending $x$ to $gxg^{-1}$ is given by conjugation with $g$. Certainly a set invariant under conjugation spans a normal subgroup. Certainly conjugacy classes are not subgroups themselves, they don't need to contain the neutral element. $\endgroup$
    – HenrikRüping
    Commented Jul 20, 2020 at 8:07
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    $\begingroup$ @R. Srivastava Rajeev welcome to MO. Maybe best if you specify where you have read this. This might make it easier to give you an answer $\endgroup$
    – Claus
    Commented Jul 20, 2020 at 8:19

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Regarding your first question, I think the comment of Henrik Rüping gives the best answer.

Regarding your second question, I am not sure about a geometric interpretation, but maybe the following perspective is a helpful starting point for your intuition: look at the center of the group.

Let's call your group $G$ and define its center $Z(G)$ by $Z(G):=\{g\in G \mid \forall x\in G: x=g^{-1}xg\}$. In other words, it is the set of elements for which the conjugacy class of each element is the element itself.

Now there are two special cases:

(1) The center is equal to $G$ iff $G$ is abelian, and in this case all conjugacy classes are just the singletons $\{x\}$ for all $x\in G$.

(2) The other extreme is if the center is trivial (i.e. just includes the identity as the only element). Examples for this are the symmetric groups $S_n$ $(n\geq 3)$ and the alternating groups $A_n$ $(n\geq 4)$.

In this sense, the order of the center measures the "degree of commutativity" of $G$ in some way, if this could be a helpful interpretation for you.

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  • $\begingroup$ +1 for a friendly constructive answer to help someone who is new on MO $\endgroup$
    – Mary Sp.
    Commented Jul 21, 2020 at 19:07

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